We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range $p\geq 2$, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.
For every $1<p\le\infty$, we prove dimension-free maximal inequalities over all radii for normalized averages over Euclidean balls in $\mathbb Z^d$. In particular, this settles the $\ell^2$ question attributed to Stein. The proof uses a two-saddle expansion at integer squared radii to compare ball multipliers with norm...
In this short note, we establish dimension-free $\ell^p(\mathbb Z^d)$ bounds, for all $p\in(1,\infty]$, for the discrete Hardy--Littlewood maximal functions associated with cubes in $\mathbb Z^d$, answering a question that had been open for a while. The key idea is to prove dimension-free bounds for the $\ell^p(\mathbb...
Mariusz Mirek, Tomasz Z. Szarek, Bla.zej Wr'obel· 0 citations
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In parti...
For $d\ge3$, let $\lambda_d$ denote the sharp constant in the nearest-neighbor Hardy inequality on $\mathbb Z^d$ with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient \[ C_d=\frac{(d-2)^2}{4} \] is sharp in dimensions three and four, wh...
Let $\Omega\subset\mathbb C^n$ be a smoothly bounded convex domain of finite type. We characterize absolute summability on its Bergman spaces and estimate approximation errors in the summing norm. For $1<p<\infty$ and $1\le q,r<\infty$, we characterize absolutely $r$-summing Carleson embeddings $A^p(\Omega)\to L^q(\mu)...
We investigate the behavior of lattice points in 1-symmetric convex bodies--those invariant under both coordinate permutations and sign changes. In this setting we introduce a discrete analogue of the isotropic constant and establish concentration of mass properties for lattice points that parallel classical mass conce...
Jakub Niksiński· 4 citations· ⚡1
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